A "normed division algebra" is an algebraic gadget where you
can add, multiply, subtract, and divide, satisfying all the usual laws
except the commutative and associative laws for multiplication,
and where every element has an "absolute value" or
"norm" satisfying the usual rules, including most notably:

|xy| = |x| |y|

The most popular example is the real numbers. The second most popular
example is the complex numbers. Then comes the quaternions, which are
noncommutative... and then, trailing in a distant fourth place,
comes the octonions, which are noncommutative and
nonassociative.

Our paper aims to give a clear and self-contained treatment of the
amazing relation between normed division algebras and supersymmetric
Yang-Mills theory. Let me explain the basic idea! I'll cut some
corners, but you can see all the details in our paper.

Suppose K is a normed division algebra of dimension n. There
are just four choices:

K = R, the real numbers, with n = 1.

K = C, the complex numbers, with n = 2.

K = H, the quaternions, with n = 4.

K = O, the octonions, with n = 8.

We get all of these by taking the real numbers and throwing in square
roots of minus 1. So, any guy in K has a "real part" and an
"imaginary part" - and we can "conjugate" it by
switching the sign of its imaginary part.

This means we can talk about hermitian matrices with entries
in K: that is, matrices that stay the same when you transpose
them and then conjugate each entry. Let's use h2(K) to mean
the set of hermitian 2×2 matrices with entries in K.

Then a nice thing happens: h2(K) is the same as (n+2)-dimensional
Minkowski spacetime! To see this, note that any guy in h2(K) has
this form:

A = t+x y
y* t-x

where t and x are real elements of K, and y is an arbitrary element.
Since K has dimension n, h2(K) has dimension n+2. And check out
its determinant:

det(A) = t2 - x2 - yy*

Note that yy* = y*y = |y|2, just as in the complex numbers.
So, det(A) is a Minkowski metric with one positive or
"timelike" direction, namely t, and n+1 negative or
"spacelike" directions, namely x and the n components of y.

So:

h2(R) is 3-dimensional Minkowski spacetime.

h2(C) is 4-dimensional Minkowski spacetime.

h2(H) is 6-dimensional Minkowski spacetime.

h2(O) is 10-dimensional Minkowski spacetime.

And - lo and behold! - these are just the dimensions where classical
superstring theory and super-Yang-Mills theory work best!

More precisely, these are the dimensions where you can write down
the Lagrangian for the "Green-Schwarz superstring" and "pure
super-Yang-Mills theory". There are fancier tricks that give
superstring theories and super-Yang-Mills theories in other
dimensions, but these are mainly offshoots of the four cases listed
here.

So now we have to ask: why do these supersymmetric theories feel so
happy when spacetime is secretly h2(K)?

Well, supersymmetry is a kind of symmetry that mixes bosons and
fermions. In the simple cases I'm talking about, this means mixing
vectors and spinors. Since vectors are the same as points in
Minkowski spacetime - once we pick an origin - vectors in dimensions
3, 4, 6, or 10 are nicely described by elements of h2(K).
And it turns out that supersymmetry works well in these dimensions
because we can also describe spinors using K. A spinor consists
of 2 guys in K: in other words, an element of K2.

Indeed, if you've studied physics, you may know that in 4d Minkowski
spacetime, where we apparently live, we use C2 to describe
spinors. I talked quite a bit about this example and also the example
of 3d Minkowski spacetime back in "week196". So go there if you want more
details. For now what matters is this:

R2 is the space of spinors in 3-dimensional Minkowski spacetime.

C2 is the space of spinors in 4-dimensional Minkowski spacetime.

H2 is the space of spinors in 6-dimensional Minkowski spacetime.

O2 is the space of spinors in 10-dimensional Minkowski spacetime.

This is a bit oversimplified, because physicists use various kinds of
spinors, and I'm not saying which kinds show up here. But I explained
all these kinds back in "week93", and I don't want to distract you with
that here. I'll say more about it later.

Now, from what we've seen so far, there's an obvious way to take a
vector and a spinor and get a new spinor. This is what matrices were
born for! Just take your matrix in h2(K), multiply your spinor in
K2 by that matrix, and you get a new spinor in K2.

In fact, we see this process at work whenever an electron absorbs
a photon. Quite literally, we see it - because that's how our eyes
work! A photon is described by a vector, an electron is described
by a spinor, and part of the math involved when an electron absorbs
a photon is this business of matrix multiplication.

Physicists would draw this operation using a Feynman diagram where a
wiggly line (the vector) and a straight one (the spinor) come in and
a straight one goes out:

Mathematicians would write it as the operation that takes A in
h2(K) and ψ in K2 and multiplies them to get
Aψ in h2(K).

Now, one cool thing about Feynman diagrams is that you can turn
them around and read them in different ways, and they still make
sense. So as soon as we have a process where a spinor absorbs a vector,
we also get a process where two spinors collide and form a vector:

Now what is this process, mathematically speaking? Well, it's some
operation that takes two spinors, say ψ in K2 and φ
in K2, and creates a vector in h2(K) that I'll call
ψ·φ. If you want the explicit formula for this
operation, read our paper.

So, we've got an operation that takes a vector and a spinor and
creates a spinor, and an operation that takes two spinors and creates
a vector. Actually, these operations exist for any dimension of
spacetime! In general we need to describe them using the language of
Clifford algebras. Only in dimensions 3, 4, 6 and 10 can we describe
them using the language of normed division algebras, as I've done
here.

But it's this special language that gives the prettiest explanation of
a certain astounding fact. Supersymmetry for the Green-Schwarz
superstring and pure super-Yang-Mills theory relies on a special
identity which is true only in dimensions 3, 4, 6 and 10:

(ψ·ψ) ψ = 0

This is an example of a "Fierz identity". These equations
show up whenever you work with spinors, and they should probably be
called "fierce identities", because they can be pretty
scary. In particular, it's a bit scary how some of them - like this
one - hold only in certain special dimensions.

But this particular one has a beautiful proof in terms of normed
division algebras! It follows from a special property shared by these
algebras: they're all "alternative". In other words, the
"associator"

[x,y,z] = (xy)z - x(yz)

changes sign whenever we switch any of the two variables. The
associator is just zero for R, C, and H, since these algebras are
associative. So the only really interesting case is the octonions,
which are not associative, but still alternative. And this is
the case that matters for superstring theory in 10 dimensions!

Anyway, what our paper does is describe the basic operations involving
spinors and vectors using the normed division algebras, then use
this description to prove the identity

(ψ·ψ) ψ = 0,

and then explain how this identity is crucial in supersymmetric
Yang-Mills theory. None of this is particularly new! What's new, we
hope, is that we explain everything in one place, in a way that people
who don't know about division algebras or supersymmetry can follow.
Some of the proofs use a little Clifford algebra technology, but most
of them amount to simple calculations.

Now let me tell you a tiny bit about the history of this subject, with
references. I would love to hear more details from people who were
around at the time. As far as I can tell, this is the first paper that
explained super-Yang-Mills theory and why it only works in dimensions
3, 4, 6 and 10:

Back in 1983, Kugo and Townsend showed how spinors in dimension 3, 4,
6, and 10 get special properties from the normed division algebras.
They formulated a supersymmetric model in 6 dimensions using the
quaternions, and speculated about a similar formalism in 10 dimensions
using the octonions:

13) Jörg Schray, The general classical solution of the superparticle,
Class. Quant. Grav. 13 (1996), 27-38. Also available as arXiv:hep-th/9407045.

They also write:

It is well-known that the 3-Ψ's rule holds for Majorana spinors in
3 dimensions, Majorana or Weyl spinors in 4 dimensions, Weyl spinors
in 6 dimensions and Majorana-Weyl spinors in 10 dimensions. Thus,
the Green-Schwarz superstring exists only in those cases. As was
shown by Fairlie and Manogue, the 3-Ψ's rule in all these cases
is equivalent to an identity on the gamma-matrices, which holds
automatically for the natural representation of the gamma-matrices
in terms of the 4 division algebras R, C, H and O, corresponding
precisely to the above 4 types of spinors. Manogue and Sudbery
then showed how to rewrite these spinor expressions in terms of 2x2
matrices over the appropriate division algebra, thus eliminating
the gamma-matrices completely.

Now I feel like explaining all this Majorana/Weyl business a bit
better - leaving many details to "week93".

First I should admit, for the nonexperts, that I've committed a few
sins of oversimplification for the sake of a nice clean story line.
For starters, remember how I said that the absorption of a photon by
an electron:

corresponds to the operation where we take a guy in h2(K)
and a guy in K2 and multiply them to get a guy in
h2(K)?

In saying this, I was ignoring everything about energy and momentum,
and focusing on the "spin" aspect of this absorption process. It's
only the "spin", or intrinsic angular momentum, of a photon that's
described by an element of h2(K) - with K = C, since we live in
4-dimensional spacetime. And it's only the spin of the electron
that's described by an element of K2.

But it's even worse than that. In 4-dimensional spacetime, spinors
come in left- and right-handed forms. For example, the neutrinos we
most easily see - not that easily, actually! - are left-handed
spinors, while antineutrinos are right-handed. Electrons come in both
left and right-handed forms, so we actually describe them using
C2 ⊕ C2 = C4. We call C4
the space of "Dirac spinors", and we call the two pieces
left- and right-handed "Weyl spinors".

Similar but subtly different things happen in other dimensions. As
far as our division algebras story goes, the crucial fact is that
besides the "obvious" way for an element of h2(K)
to act on K2, there is a less obvious way that involves the
"traced-reversed" form of a 2×2 hermitian matrix:

A~ = A - tr(A)

where the trace tr(A) is the sum of the diagonal entries. We get
one kind of spinors, say

S+ = K2

upon which the vectors

V = h2(K)

act in the obvious way, and another kind of spinors, say

S- = K2

in which vectors act in a nonobvious way. As vector spaces S+ and
S- are the same - but they differ in how vectors act on them, and we
should think of this action as interchanging the two kinds of spinor.
Here are the formulas:

V ⊗ S+ → S-
A ⊗ ψ |→ Aψ

V ⊗ S- → S+
A ⊗ ψ |→ A~ ψ

These actions fit together to yield a Clifford algebra action on
the direct sum of S+ and S-, since

A A~ = A~ A = -det(A)

and the determinant is related to the metric on Minkowski spacetime,
so these are the Clifford algebra relations in deep disguise.

What all this really amounts to depends a lot on which of the four
division algebras we're looking at! Sometimes S+ and
S- are secretly isomorphic, sometimes not. They always
start out being real vector spaces, since as vector spaces they're
just K2, and the only uniform way to think of all four
normed division algebras is as real vector spaces. But sometimes
S+ and S- admit Lorentz-invariant complex
structures, so we can think of them as complex vector spaces!

(By "Lorentz-invariant" I really mean invariant under the action of
the double cover of the Lorentz group. For brevity, let's just call
this the Lorentz group.)

In fact, each of the four cases has its own unique personality, with
the 4d case being the weirdest - you might call it a "split
personality". Let me just summarize the facts, without much
explanation. This is one of those things where I write stuff down so
I can forget it and look it up later:

If K = R, we're in 3d Minkowski spacetime. Then S+ and
S- are isomorphic as real representations of the Lorentz
group - so it's not important to distinguish them. The secret reason
for this is that R is commutative. Since S+ ≅
S- does not have an invariant complex structure, we call
elements of this space "Majorana" spinors, which is the name
for real spinors that don't have a particular handedness.

If K = C, we're in 4d Minkowski spacetime. Then S+ and
S- are isomorphic as real representations of the Lorentz
group - so it's not important to distinguish them. The secret reason
for this is that C is commutative. If we treat S+ ≅
S- as a real vector space, we call elements of this space
"Majorana" spinors, which is the name for real spinors that
don't have a particular handedness.

But in fact this real vector
space has two invariant complex structures, and the resulting complex
representations are not isomorphic! If we think of S+
and S- as two nonisomorphic complex representations we call
their elements left- and right-handed "Weyl" spinors,
respectively - since that's the name for complex spinors that do have
a particular handedness.

If K = H, we're in 6d Minkowski spacetime. Then S+ and
S- are not isomorphic as real representations of the
Lorentz group - so it's important to distinguish them. The secret
reason for this is that H is not commutative. Furthermore,
S+ and S- admit invariant complex structures.
If we think of S+ and S- as complex
representations we call their elements left- and right-handed
"Weyl" spinors, respectively - since that's the name for
complex spinors that do have a particular handedness.

If K = O we're in 10d Minkowski spacetime. Then S+ and
S- are not isomorphic as real representations of the
Lorentz group - so it's important to distinguish them. The secret
reason for this is that O is not commutative. Furthermore,
S+ and S- do not admit invariant complex
structures. So, we must think of S+ and S- as
real representations, and we call their elements left- and
right-handed "Majorana-Weyl" spinors, respectively - since
that's the name for real spinors that do have a particular handedness.

Wow, I bet that was thrilling!

Now that I'm done with this paper, my life has undergone a phase
change. I've been finishing a lot of old papers for the last 2 years.
This is the last of that batch - and the least old. It comes as an
incredible relief. Working on old projects is tiring, especially when
you have new things you're dying to think about. I feel like I've
been way behind myself, running to catch up... but this week I finally
caught up and ran past myself! It's a strange sensation.

Adding to this strange sensation, I just got word that I'm free to
take leave from UCR and visit Singapore for a year, starting in July
2010. I'll be working at the Centre for Quantum Technologies. That
should be a great adventure.

So, I'm feeling peppy, and I'm dying to tell you about all sorts of
new cool stuff: Stirling numbers and the Poisson operad, stacks and
noncommutative geometry, Adams operations and Galois theory, toric
varieties, octonions and rolling balls, the windmill powered by light,
and the symplectic geometry of electrical circuits. Each of these
deserves a whole Week, but we'll see.

For now, here are a few cool things I won't tell you much about,
because I don't understand them well enough. First, as pointed out to
me by Mike Stay:

This starts with the old problem of trying to find a number x such
that xn = 2 and xk is almost 3/2. In music
jargon, this is called "finding an equal tempered scale that has
a good fifth". In math jargon, it amounts to finding a good
rational approximation to

In relation with your recent, interesting, arxiv:0909.0551 paper I
would like to signal that division algebras also appear in the
N-extended supersymmetric quantum mechanics (in one dimension) for
N=1,2,4,8. This is hardly surprising, of course (arXiv:hep-th/0109073 NPB
Pr. Sup.). Perhaps slightly more surprising is the fact that the
octonionic structure constants enter, as coupling constants, N=8
invariant actions, like e.g. the (1,8,7) model of arXiv:hep-th/0511274 (also
in JHEP). In this example the 7 auxiliary fields can be associated
with the 7 imaginary octonions, preserving the "octonionic
covariance". I should add that the representations of N-extended 1D
superalgebra are mathematically very interesting and quite intricate.
In the last few years basically two groups, my group and the group of
S. Gates and his collaborators, worked out with complementary
viewpoints and results several features of these representations:
total number of fields, grading of the fields, graph interpretation,
connectivity of the graphs, etc. etc. Basic references can be found
in hep-th/0010135
(in JMP), hep-th/0610180 (PoS), or
typing my name (and Gates'name) in the arXiv. Perhaps you could find
these interesting to have a look at.